There are several gameplay differences between these versions; the principal ones are summarized below:

Crashes when landing from jumps happen very often in BB 1.0 and MS 1.1 (1990 one), as the maximum angle allowed for hitting the ground without crashing was much smaller.

Unlike on the various 1.1 versions, with BB 1.0 it is generally not possible to gain speed when landing from jumps - as you hit the ground, the horizontal velocity of the car will remain unchanged. That makes an enormous difference in car behaviour, specially for the slow cars, and also means it becomes virtually impossible to reach powergear with Ferrari and Corvette.

Some track elements characteristics also change. For instance, tunnel roofs are completely penetrable in BB 1.0, and while it is possible to land on them with MS 1.1 versions, you will often be spared from a crash when hitting them from below.

There are also minor performance and handling differences. Overall, cars are slowest in BB 1.0 and fastest in BB 1.1.

All these differences cause replays performed with different game versions incompatible regardless of the tracks being exchangeable, as the game engine will process the input stored in the replay file differently for each version.

The most used version in competitions nowadays is Stunts BB 1.1. At the dawn of Stunts competitions (1998-2000), however, it was Stunts BB 1.0 that was most popular. When the user share of Stunts BB 1.1 began to grow, a cheated car named Contest Car was created by Lukas Loehrer to allow Stunts 1.0 users be as fast as 1.1 users, because Indy car is slightly faster in BB/MS 1.1 versions than in BB 1.0 version.

According to the countries, Mindscape or Broderbund had the contract for selling the game. In USA, Germany and Eastern Europe the publisher was Broderbund; in France, U.K. and Belgium it was Mindscape. Overall, Broderbund versions are much more well known worldwide.

Amiga version

It seems that only publisher for Amiga was Mindscape; therefore, the game was published under the name 4D Sports Driving.
Only one version was available : it was version was 1.2, converted to Amiga from PC version.
This version has less bugs than PC versions.

FM Towns version

FM Towns/Marty computers were i386 CPU-based computers NOT compatible with classic IBM-compatible PCs, made by Fujitsu (FM meaning Fujitsu Micro) and sold only in Japan. A console FM Towns, compatible with FM Towns/Marty computers, was also created.

The editor for this version was Electronic Arts (E.A.) - Victor, as Distinctive Software has been bought by E.A. in 1991 to become E.A. Canada.
This version was also converted from PC, and with less bugs than PC versions.

As name "Stunts" was property of Brøderbund and "4D Sports" name was property of Mindscape, the game was released as "4D Driving" (to remember 4D Sports - Driving name that was already known in Japan due to Amiga version of the game).

Box, manual and CD box of FM Towns version 1.0

Rear view of FM Towns box

PC-9801 version

Title screen of PC-9801 version 1.0

Another Japanese port was made for the NEC PC-9801. It was produced by the same team as the FM Towns version and the two ports shared graphics and new opponents. Due to hardware limitations this version had fewer colors and the music were synthesized. Otherwise, it is quite similar to the FM Towns version.

Other versions

Kevin Pickell mentioned, in an Wikipedia edit of the Stunts article, that a Sega Genesis (AKA Mega Drive) version was planned and realized, but the port never made it to the stores as it was not possible to make it work at an acceptable frame rate.